Astron. Astrophys. 356, 127-133 (2000)

3. Extra-tidal excess in M 92

3.1. Radial density profile

As first step, we built a 2-D star density map by binning the catalog in step of . Then, we fitted a polynomial surface to the background, selecting only the outermost regions of the studied area. The background correction is expressed in the same units of 2-D surface density counts, and can be direclty applied to the raw counts. A tilted plane was sufficient to interpolate the background star counts (SCs). Higher-order polynomials did not provide any substantial improvement over the adopted solution. We compared the fitted background with IRAS maps at 100µ, but we did not find any direct signs of a correlation between the two. Rather, the direction of the tilt is consistent with the direction of the galactic center. Hence, the tilt of the background, which is however very small (), can be considered as due to the galactic gradient.

The cluster radial density profile was obtained from the background subtracted SC's by counting stars in annuli of equal logarithmic steps. The uncorrected surface density profile (hereafter SDP) is expressed as:

where indicates the number of objects in the annulus between and , and the area of the annulus. The constant was determined by matching the profile with the published profile of T95 in the overlap range. The effective radius at each point of the profile is given by:

The SDP must now be corrected for crowding. When dealing with photographic material, it is not possible to apply the widely known artificial star technique used with CCD data. Therefore, we used a procedure similar to the ones described in Lehman & Scholz (1997) and Garilli et al. (1999): we estimated the area occupied by the objects in each radial annulus by selecting all the pixels brighter than the background noise level plus three sigmas, and considered as virtually uncrowded the external annuli in which the percentage (very small, ) of filled area did not vary with the distance. The external region starts at from the cluster center with a filling factor smaller than . After correcting the area covered by non-stellar objects, the ratio unfilled/filled area gives the crowding correction. This correction was computed at the effective radii of the surface brightness profile (hereafter SBP) and smoothed using a spline function. The corrected surface brightness profile was then computed as:

where frac is the crowding correction factor determined at the i point on the profile.

The crowding corrected SBP of M 92 derived from DPOSS data is shown in Fig. 4 as filled dots and the uncorrected counts as crosses. Table 1 lists the measured surface brightness profile. With a simple number counts normalization we joined our profile to the one (open circles) derived by T95, in order to extend the profile to the inner regions. We then fitted a single-mass King model to our profile. The fitting profile is drawn on Fig. 4 as a continuous line. Our value for the tidal radius, , turned out to be similar to the value given in Brosche et al. (1999), and slightly smaller than the one given in T95, . As it can be seen from the figure, DPOSS data extend at larger radial distances than the T95 compilation and reveal the existence of a noticeable deviation from the isotropic King model derived from the direct fitting of the SBP. This deviation is a clear sign of the presence of extra tidal material. We also tried fitting anisotropic King models to the SBP, but the fit was not as good as in the isotropic case.

Fig. 4. M 92 radial density profile. Open dots, Trager et al. (1995) data; filled dots, DPOSS SCs; crosses, crowding uncorrected SCs; solid line, isotropic King model.

Table 1. Measured surface brightness profile

At what level is this deviation significant? The determination of the tidal radius of a cluster is still a moot case. While fitting a King model to a cluster density profile, the determination of the tidal radius comes from a procedure where the overall profile is considered, and internal points weigh more than external ones. On the one hand, this is an advantage since the population near the limiting radius is a mix of bound stars and stars on the verge of being stripped from the cluster by the Galaxy tidal potential. On the other hand, the tidal radius obtained in this way can be a poor approximation of the real one. In the classical picture, and in presence of negligible diffusion, the cluster is truncated at its tidal radius at perigalacticon (see Aguilar et al. 1988). Nevertheless, Lee & Ostriker (1987) pointed out that mass loss is not instantaneous at the tidal radius, and, for a given tidal field, they expect a globular cluster to be more populated than in the corresponding King model. Moreover, a globular cluster along its orbit also suffers from dynamical shocks, due to the crossing of the Galaxy disk and, in case of eccentric orbits, to close passages near the bulge, giving rise to enhanced mass-loss and, later on, to the destruction of the globular cluster itself. Gnedin & Ostriker (1997) found that, after a gravitational shock, the cluster expands as a whole, as a consequence of internal heating. In this case, some stars move beyond the tidal radius but are not necessarily lost, and are still gravitationally bound to the cluster. This could explain observed tidal radii larger than expected for orbits with a small value of the perigalacticon. Brosche et al. (1999) point out that the observed limiting radii are too large to be compatible with perigalacticon , and suggest that the appropriate quantity to be considered is a proper average of instantaneous tidal radii along the orbit. It can be seen from Fig. 4 that the cluster profile deviates from the superimposed King model before the estimated tidal radius, and has a break in the slope at about , after which the slope is constant. We shall come back later to this point.

In Fig. 5 we show the surface density profile, expressed in number of stars to allow a direct comparison with J99, and binned in order to smooth out oscillations in the profile, due to the small S/N ratio arising with small-sized annuli. J99 predict that stars stripped from a cluster, and forming a tidal stream, show a density profile described by a power law with exponent . We fitted a power law of the type to the extra-tidal profile (dashed line). The best fit gives a value and is shown as a dashed line in Fig. 5. The errors on the profile points include also the background uncertainty, in quadrature, so that the significance of the extra-tidal profile has been estimated in terms of the difference , where is the number surface density profile at point i, and its error, which includes the background and the signal Poissonian uncertainties. This quantity is positive for all the points except for the outermost one. The fitted slope is consistent with the value proposed by J99 and in good accordance with literature values for other clusters (see G95 and Zaggia et al. 1997).

Fig. 5. Fit of a power law to the external profile of M 92, expressed in number surface density to allow for an immediate comparison with Johnston et al. (1999, hereafter J99), and binned to smooth out oscillations due to the small dimension of the annuli. Triangles indicate the profile of the extra-tidal halo. Diamonds represent the binned and averaged profile within the tidal radius. The dashed line is the fitting power law with .

We then fit an ellipse to the extra-tidal profile in order to derive a position angle of the tidal extension, and checked whether the profile in that direction differs from the one obtained along the minor axis of the fitting ellipse, to confirm that the extra-tidal material is a tail rather than a halo. The best fitting ellipse, made on the "isophote" at the level from the background (approximately from the center of the cluster), turned out to have a very low ellipticity, ( at P.A.). We have also measured the radial profiles along the major and minor axes, using an aperture angle of , in order to enhance the S/N ratio in the counts. The two profiles turned out to be indistinguishable within our uncertainties. This result shows that the halo material has a significantly different shape than the internal part of the cluster which shows an ellipticity of at P.A. as found by White & Shawl (1987).

3.2. Surface density map

In the attempt to shed more light upon presence and characteristics of the extra-tidal extension, we used the 2-D star counts map, as described at the beginning of the previous section. We applied a Gaussian smoothing algorithm to the map, in order to enhance the low spatial frequencies and cut out the high frequency spatial variations, which contribute strongly to the noise. We smoothed the map using a Gaussian kernel of . The resulting smoothed surface density map is shown in Fig. 6. Since the background absolute level is zero, the darkest gray levels indicate negative star counts. In this image, the probable tidal tail of M 92 (light-gray pixels around the cluster) is less prominent than in the radial density profile: this is because data are not averaged in azimuth. On the map we have drawn three "isophotal" contours at 1, 2 and 3 over the background. The fitted tidal radius is marked as a thick circle and the two arrows point toward the galactic center (long one) and in the direction of the measured proper motion (see Dinescu et al. 1999). The tidal halo does not seem to have a preferred direction. A marginal sign of elongation is possibly visible along a direction almost orthogonal to that of the galactic center.

Fig. 6. M 92 surface density map from background subtracted star counts. The black, thick circle is drawn at the estimated tidal radius of M 92. The long, thicker arrow indicates the direction of the galactic center, the thin arrow indicates the proper motion of the cluster as in Dinescu et al. (1999). Contours are drawn at 1,2 and 3 of the background.

As pointed out in the previous section, if we build the profile along this direction and orthogonally to it, we do not derive clear signs of any difference in the star count profiles in one direction or the other, mainly because of the small number counts.

On the basis of these results, we can interpret the extra-tidal profile of M 92 as follows: at radii just beyond the fitted King profile tidal radius, the profile resembles a halo of stars -most likely still tied up to the cluster or in the act of being stripped away. As the latter process is not instantaneous, these stars will still be orbiting near the cluster for some time. We cannot say whether this is due to heating caused by tidal shocks, or to ordinary evaporation: a deep CCD photometry to study the mass function of extra-tidal stars would give some indications on this phenomenon. At larger radii, the 1 "isophote" shows a barely apparent elongation of the profile in the direction SW to NE, with some possible features extending approximately towards S and E. Although the significance is only at 1 level, these structures are visible and might be made up by stars escaping the cluster and forming a stream along the orbit. As pointed out in Meylan & Heggie (1997), stars escape from the cluster from the Lagrangian points situated on the vector connecting the cluster with the center of the Galaxy, thus forming a two-sided lobe, which is then twisted by the Coriolis force. A clarifying picture of this effect is given in Fig. 3 of Johnston (1998).

© European Southern Observatory (ESO) 2000

Online publication: March 28, 2000
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